The EKR-Module Property of Pseudo-Paley Graphs of Square Order

Abstract

We prove that a family of pseudo-Paley graphs of square order obtained from unions of cyclotomic classes satisfies the Erdős-Ko-Rado (EKR) module property, in a sense that the characteristic vector of each maximum clique is a linear combination of characteristic vectors of canonical cliques. This extends the EKR-module property of Paley graphs of square order and solves a problem proposed by Godsil and Meagher. Different from previous works, which heavily rely on tools from number theory, our approach is purely combinatorial in nature. The main strategy is to view these graphs as block graphs of orthogonal arrays, which is of independent interest.

Erdős–Ko–Rado theorem in Peisert-type graphs

“…Theorem 2.2 is best possible if q is a square; see the counterexamples constructed in [1, Theorem 9]. We refer to [2, Corollary 4.1] for a slightly stronger statement.…”

“…. , c m F q are maximum cliques because of the Delsarte bound (see [1,Theorem 7] for a different proof). Thus, translates of c 1 F q , c 2 F q , .…”

“…Indeed, one can read from the connection set S that are cliques in X containing . Moreover, Peisert-type graphs are based on cyclotomic constructions (due to Brouwer, Wilson, and Xiang [8]) so that they are strongly regular, which implies that are maximum cliques because of the Delsarte bound (see [1, Theorem 7] for a different proof). Thus, translates of can be regarded as canonical cliques in X .…”

“…Furthermore, given a Peisert-type graph, it seems difficult to predict if it has the strict-EKR property by staring at its connection set. Indeed, one can systematically construct an infinite family of Peisert-type graphs for which the strict-EKR property fails [1, Section 5].…”

“…In [1, Section 6], Asgarli, Goryainov, Lin, and Yip asked for a complete classification of Peisert-type graphs with the strict-EKR property. This question is potentially very challenging; indeed, it turns out to be a special case of a widely open problem regarding the strict-EKR property of block graphs of orthogonal arrays [12, Problem 16.4.1].…”